Let X_1, X_2,...,X_n ~ N(mu, sigma^2) be i.i.d random variables. Setting hat{mu}_n = (1/n) sum_{i=1}^n X_i, find the mean and variance of sqrt{n}(hat{mu}_n - mu).
Understand the Problem
The question is asking to find the mean and variance of the expression ( \sqrt{n}(\hat{\mu}_n - \mu) ), where ( \hat{\mu}_n ) is the sample mean of i.i.d. random variables that follow a normal distribution. We will use properties of the normal distribution and the behavior of sample means to solve for the mean and variance.
Answer
Mean: $0$, Variance: $\sigma^2$
Answer for screen readers
The mean of ( \sqrt{n}(\hat{\mu}_n - \mu) ) is ( 0 ) and the variance is ( \sigma^2 ).
Steps to Solve
- Identify the components of the expression
We know that ( \hat{\mu}_n ) is the sample mean of ( n ) independent and identically distributed (i.i.d.) random variables from a normal distribution. The general properties of the sample mean state:
- The mean of ( \hat{\mu}_n ) is equal to the population mean ( \mu ).
- The variance of ( \hat{\mu}_n ) is given by ( \frac{\sigma^2}{n} ), where ( \sigma^2 ) is the population variance.
- Find the mean of ( \sqrt{n}(\hat{\mu}_n - \mu) )
Using the linearity of expectation, we have: $$ E[\sqrt{n}(\hat{\mu}_n - \mu)] = \sqrt{n}(E[\hat{\mu}_n] - \mu) $$ Since ( E[\hat{\mu}_n] = \mu ): $$ E[\sqrt{n}(\hat{\mu}_n - \mu)] = \sqrt{n}(\mu - \mu) = 0 $$
- Find the variance of ( \sqrt{n}(\hat{\mu}_n - \mu) )
Using the property that the variance of a constant multiplied by a random variable is the constant squared times the variance of that variable, we can compute: $$ \text{Var}(\sqrt{n}(\hat{\mu}_n - \mu)) = n \cdot \text{Var}(\hat{\mu}_n) $$ Substituting in the expression for the variance of ( \hat{\mu}_n ): $$ \text{Var}(\sqrt{n}(\hat{\mu}_n - \mu)) = n \cdot \frac{\sigma^2}{n} = \sigma^2 $$
The mean of ( \sqrt{n}(\hat{\mu}_n - \mu) ) is ( 0 ) and the variance is ( \sigma^2 ).
More Information
This result is significant as it aligns with the central limit theorem, which states that as the sample size ( n ) increases, the distribution of the sample mean approaches a normal distribution centered around the population mean, with a reduced spread characterized by the variance decreasing with ( n ).
Tips
- A common mistake is to overlook the property of the mean of ( \hat{\mu}_n ), mistakenly assuming it equals ( 0 ) instead of ( \mu ).
- Another error is incorrectly applying the variance properties; remember that multiplying by a constant scales the variance by the square of that constant.
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